In recent years, studies are being actively carried out on quantum well structures having physical properties different from those of general crystal materials. For example, superlattice semiconductors including laminated semiconductor thin-films exhibit a band structure that varies depending on the period (thickness) of layers and type of atoms, and therefore applications to a variety of devices is expected. In a quantum dot which is a three-dimensional quantum confinement structure, a density of states is discretized and concentration occurs on a specific state, and it is thereby theoretically possible to implement a laser medium with extremely high efficiency. Physical properties of such a quantum well structure fluctuate a great deal depending on a period of crystal lattice and a degree of disorder or the like, and so it is necessary to acquire accurate information on the crystal lattice when evaluating those properties.
To evaluate regularity of the crystal lattice, a method is used which irradiates a sample with X-rays and measures a scattering intensity distribution in a reciprocal space. According to this method which is also called “reciprocal space mapping” or “reciprocal space map,” a scattering intensity of X-rays in the vicinity of reciprocal lattice points is mapped and a scattering power distribution is obtained. With an ideal crystal structure, the diffraction intensity of X-rays becomes stronger only at reciprocal lattice points, but when there is disorder in a crystal lattice, significant scattering power is observed also at positions away from reciprocal lattice points.
The basic concept of reciprocal space mapping will be described. First, a relationship between a real space and a reciprocal space will be described in brief. FIG. 6 is a schematic view illustrating a relationship between a real space and a reciprocal space in a two-dimensional crystal. FIG. 6A shows a crystal lattice made up of atoms A1 to A4. In this crystal lattice, for example, the lattice spacing of a 100-plane is d100, the lattice spacing of a 010-plane is d010 and the lattice spacing of a 110-plane is duo.
When the crystal lattice in FIG. 6A is transformed into a reciprocal space, the transformed crystal lattice appears as shown in FIG. 6B. The reciprocal space corresponds to a Fourier transform of a real space and a reciprocal lattice point includes information of the crystal lattice in the real space. For example, as shown in FIG. 6B, the distance between an origin O of the reciprocal space and a certain reciprocal lattice point corresponds to a reciprocal of the lattice spacing of the corresponding crystal plane. To be more specific, the distance between the origin O of the reciprocal space and the point 100 corresponds to the reciprocal 1/d100 of the lattice spacing d100 of the 100-plane, and the distance between the origin O of the reciprocal space and the point 010 corresponds to the reciprocal 1/d010 of the lattice spacing d010 of the 010-plane.
FIG. 7 is a schematic view illustrating a relationship between a reciprocal lattice point and an Ewald sphere. FIG. 7 illustrates a reciprocal space corresponding to a three-dimensional crystal when seen from a qy-axis (not shown) direction perpendicular to the plane including a qx-axis and a qz-axis. In FIG. 7, a K0 vector represents a wave number vector of incident X-rays incident upon the crystal structure and a K vector represents a wave number vector of scattered X-rays scattered by the crystal structure. In FIG. 7, a plurality of regularly arranged white circle marks represent reciprocal lattice points.
The scattering intensity of X-rays by an ideal crystal structure without crystal lattice disorder becomes stronger under a condition that a reciprocal lattice point exists on a spherical surface of the Ewald sphere E0 having a radius (2π/λ), 2π times the reciprocal of wavelength (λ) of X-rays. In this case, X-rays are scattered in a projected pattern obtained by projecting reciprocal lattice points located on the spherical surface of the Ewald sphere E0 from the center of the Ewald sphere E0. When there is crystal lattice disorder, the scattering intensity becomes stronger also at points other than the reciprocal lattice points according to the degree of the disorder. For this reason, it is possible to evaluate regularity of the crystal lattice by calculating a scattering intensity distribution through reciprocal lattice mapping.
To be more specific, for example, when a plurality of crystal lattices having different intervals are mixed in a sample and reciprocal lattice points on the qz-axis are located on the Ewald sphere, scattering power appears along a straight line connecting the origin and the reciprocal lattice points (qz-axis direction in FIG. 7). On the other hand, when a plurality of crystal planes (lattice planes) having different inclinations are mixed in a sample, scattering power appears in a direction orthogonal to a straight line connecting the origin and a reciprocal lattice point in the reciprocal space (q′x-axis direction in FIG. 7). Thus, by checking the distribution of scattering power, it is possible to evaluate regularity of the crystal lattice.
In reciprocal space mapping, a scattering intensity distribution of X-rays in the vicinity of a target reciprocal lattice point is normally measured. FIG. 8 is a schematic view illustrating an example of a measuring apparatus used for reciprocal lattice mapping. In a measuring apparatus 2 shown in FIG. 8, -monochromatic (single wavelength) X-rays that are emitted from an X-ray source 201 and passed through a monochromator 202 is incident on a sample 203 at a glancing angle (complementary angle of an angle of incidence) ω. X-rays scattered at the sample 203 is incident on a detector 205 via a collimator 204. The collimator 204 selectively guides only X-rays scattered toward a 20 direction from the sample to the detector 205.
An example of the measuring mode of reciprocal space mapping using the measuring apparatus 2 is ω scanning. In ω scanning, ω is changed with 2θ fixed to a predetermined value and a scattering intensity distribution in the ω direction is scanned. Through this scanning in the ω direction, a scattering power distribution in the direction substantially orthogonal to the straight line connecting the origin and the reciprocal lattice point in the reciprocal space (q′x-axis direction in FIG. 7) is detected. When scanning in the ω direction for a certain 2θ is finished, the value of 2θ is slightly changed and scanning in the ω direction is performed again. Selecting a different value of 2θ has a meaning equivalent to selecting a different qz position along the qz-axis of the reciprocal space. In this way, by repeating scanning in the ω direction every time the value of 2θ is changed, it is possible to obtain a scattering power distribution in a region RSM within the qx-qz plane, that is, a two-dimensional scattering intensity distribution corresponding to a reciprocal space map.
Another example of the measuring mode is ω-2θ scanning. In this measuring mode, a scattering intensity distribution is scanned so that the amount of change Δω of ω and the amount of change Δ(2θ) of 2θ always satisfy a relationship of Δω:Δ(2θ)=1:2. Furthermore, the above scanning is repeated every time the initial value ω0 of ω is changed. Scanning that satisfies Δω:Δ(2θ)=1:2 corresponds to measuring a scattering power distribution in a straight line direction that passes through a given point of the reciprocal space and the origin. Changing ω0 corresponds to specifying a different position on the q′x-axis. Thus, by repeating ω-2θ scanning every time ω0 is changed, it is possible to measure the scattering power distribution within the region RSM in FIG. 7.
FIG. 9 is a schematic view illustrating another example of the measuring apparatus used for reciprocal space mapping. In a measuring apparatus 3 shown in FIG. 9, monochromatic X-rays that are emitted from an X-ray source 301 and passed through a monochrometor 302 enter a sample 303 at a glancing angle ω. X-rays scattered at the sample 303 is incident on a one-dimensional detector 305. The one-dimensional detector 305 is configured so as to be able to measure a wide scattering angle (2θ direction) simultaneously. For this reason, scanning in the 2θ direction is not necessary in reciprocal space mapping using the measuring apparatus 3.